Exercise 3: Convergent Sequence inside a Continuous Function

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description

In this video I prove Theorem 3, which states that a convergent sequence with limit L can be plugged directly into a continuous function at L, thus obtaining f(L). I prove this using the precise definition of limits of general functions and sequences. This theorem is also expected since a function at a particular value is equal to the limit as x approaches that particular value; thus we are just replacing the x term with the infinite sequence.

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Timestamps:

Exercise 3: Theorem 3: Sequence inside a function: 0:00
Solution using the precise definition of limits: 1:27
f is continuous at L so limit is same as plugging in function: 3:45
Replace precise definition of a limit for a continuous function with the sequence: 5:54
Precise definition of the limit is the same for our sequence: 7:29

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